In Russian we also would ask "What's animal is it?" (Что это за зверь?). But the question was about the phrase without the word "animal", because these are 2 different constructions (at least in Russian).

It can never be vem (who). (Unless possibly in a situation in which the speaker knows a group of animals ("by name") and asks who (of them) is it? Obviously this requires the animals in question to be so familiar that they have gone from being animals to individuals.)

I'm not sure about patterns, but in Russian animals are grammatically treated as animate objects. I was actually thinking how far "down the evolution" it goes and I think it stops somewhere at insects, e.g. a cat is a she (Russian has grammatical genders), a cockroach either an it or a he and an amoeba an it.

So, if an animal destroyed a nest and we would like to know whether it was a raccoon, a fox or a cat, in Russian we would ask "who destroyed the nest?” (which in English sounds comical as I learned the hard way ), and in English we would say “what destroyed the nest?”.

Obviously in English we say “who” in certain situations, e.g. when I come to take horse riding lessons, I ask my instructor “who I am riding today?”, because all the horses in the stable are known by the name, sex and personality.

Oh but that reminds me: yes, we do say 'wat' (what, indeed), but we also use the pronouns you use. In that sense they're animate too - but then our pens, cars, ships, etc., all are animate as well - and that cannot (can hardly?) be considered true, can it? The so-called het-words are generally inanimate, for sure, but not of course members and children ('het lid', 'het kind');they are certainly animate too. That would be a bridge too far. Our gender indications are mainly grammatical.
We'd use 'who' too, I guess, when it is a pet, etc.

Not really. In English there are many situations where they say he/she/who about animals, e.g. if the nest was destroyed by one of the three house cats, the questions would be "who?", but it is a very specific context. Sometimes if you know the sex of the animal from the context (a mother with babies, a deer with antlers), they say he/she.

Hebrew:
we do say what is it, but we mean in the absolute form - *what* it is.
we do take it as an animated object, and we dont have 'it' system, only male and female, so we would ask about anything what it is.
So maybe we do not differentiate between the two.

we use "who" only to ask about humans
we don't have a gender-neutral pronoun like "it" in English, we usually use "this" which can be masculine or feminine and if we don't know the gender we use the masculine form.